/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 206 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxWeightedTrs
(9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTypedWeightedTrs
(11) CompletionProof [UPPER BOUND(ID), 0 ms]
(12) CpxTypedWeightedCompleteTrs
(13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 7 ms]
(14) CpxRNTS
(15) CompleteCoflocoProof [FINISHED, 330 ms]
(16) BOUNDS(1, n^1)


----------------------------------------

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(X, g(X), Y) -> f(Y, Y, Y)
   g(b) -> c
   b -> c

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(c) -> c
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_b) -> b
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   encode_b -> b
   encode_c -> c


----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(X, g(X), Y) -> f(Y, Y, Y)
   g(b) -> c
   b -> c

The (relative) TRS S consists of the following rules:

   encArg(c) -> c
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_b) -> b
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   encode_b -> b
   encode_c -> c

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(X, g(X), Y) -> f(Y, Y, Y)
   g(b) -> c
   b -> c

The (relative) TRS S consists of the following rules:

   encArg(c) -> c
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_b) -> b
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   encode_b -> b
   encode_c -> c

Rewrite Strategy: INNERMOST
----------------------------------------

(5) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   b -> c
   f(X, c_g(X), Y) -> f(Y, Y, Y)
   g(c_b) -> c

The (relative) TRS S consists of the following rules:

   encArg(c) -> c
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_b) -> b
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   encode_b -> b
   encode_c -> c
   b -> c_b
   g(x0) -> c_g(x0)

Rewrite Strategy: INNERMOST
----------------------------------------

(7) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   b -> c [1]
   f(X, c_g(X), Y) -> f(Y, Y, Y) [1]
   g(c_b) -> c [1]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_b) -> b [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_b -> b [0]
   encode_c -> c [0]
   b -> c_b [0]
   g(x0) -> c_g(x0) [0]

Rewrite Strategy: INNERMOST
----------------------------------------

(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   b -> c [1]
   f(X, c_g(X), Y) -> f(Y, Y, Y) [1]
   g(c_b) -> c [1]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_b) -> b [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_b -> b [0]
   encode_c -> c [0]
   b -> c_b [0]
   g(x0) -> c_g(x0) [0]

The TRS has the following type information:
b :: c:c_g:c_b:cons_f:cons_g:cons_b
c :: c:c_g:c_b:cons_f:cons_g:cons_b
f :: c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b
c_g :: c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b
g :: c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b
c_b :: c:c_g:c_b:cons_f:cons_g:cons_b
encArg :: c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b
cons_f :: c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b
cons_g :: c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b
cons_b :: c:c_g:c_b:cons_f:cons_g:cons_b
encode_f :: c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b
encode_g :: c:c_g:c_b:cons_f:cons_g:cons_b -> c:c_g:c_b:cons_f:cons_g:cons_b
encode_b :: c:c_g:c_b:cons_f:cons_g:cons_b
encode_c :: c:c_g:c_b:cons_f:cons_g:cons_b

Rewrite Strategy: INNERMOST
----------------------------------------

(11) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1, v2) -> null_encode_f [0]
   encode_g(v0) -> null_encode_g [0]
   encode_b -> null_encode_b [0]
   encode_c -> null_encode_c [0]
   b -> null_b [0]
   g(v0) -> null_g [0]
   f(v0, v1, v2) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_g, null_encode_b, null_encode_c, null_b, null_g, null_f

----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   b -> c [1]
   f(X, c_g(X), Y) -> f(Y, Y, Y) [1]
   g(c_b) -> c [1]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_b) -> b [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_b -> b [0]
   encode_c -> c [0]
   b -> c_b [0]
   g(x0) -> c_g(x0) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1, v2) -> null_encode_f [0]
   encode_g(v0) -> null_encode_g [0]
   encode_b -> null_encode_b [0]
   encode_c -> null_encode_c [0]
   b -> null_b [0]
   g(v0) -> null_g [0]
   f(v0, v1, v2) -> null_f [0]

The TRS has the following type information:
b :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
c :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
f :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
c_g :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
g :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
c_b :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
encArg :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
cons_f :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
cons_g :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
cons_b :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
encode_f :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
encode_g :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f -> c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
encode_b :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
encode_c :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
null_encArg :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
null_encode_f :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
null_encode_g :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
null_encode_b :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
null_encode_c :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
null_b :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
null_g :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f
null_f :: c:c_g:c_b:cons_f:cons_g:cons_b:null_encArg:null_encode_f:null_encode_g:null_encode_b:null_encode_c:null_b:null_g:null_f

Rewrite Strategy: INNERMOST
----------------------------------------

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   c => 0
   c_b => 1
   cons_b => 2
   null_encArg => 0
   null_encode_f => 0
   null_encode_g => 0
   null_encode_b => 0
   null_encode_c => 0
   null_b => 0
   null_g => 0
   null_f => 0

----------------------------------------

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 0 }-> 1 :|: 
   b -{ 1 }-> 0 :|: 
   b -{ 0 }-> 0 :|: 
   encArg(z) -{ 0 }-> g(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> b :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_b -{ 0 }-> b :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z', z'') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_g(z) -{ 0 }-> g(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z, z', z'') -{ 1 }-> f(Y, Y, Y) :|: Y >= 0, z'' = Y, z' = 1 + X, X >= 0, z = X
   f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   g(z) -{ 1 }-> 0 :|: z = 1
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(15) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V, V2),0,[b(Out)],[]).
eq(start(V1, V, V2),0,[f(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[g(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[encArg(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[fun(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[fun1(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[fun2(Out)],[]).
eq(start(V1, V, V2),0,[fun3(Out)],[]).
eq(b(Out),1,[],[Out = 0]).
eq(f(V1, V, V2, Out),1,[f(Y1, Y1, Y1, Ret)],[Out = Ret,Y1 >= 0,V2 = Y1,V = 1 + X1,X1 >= 0,V1 = X1]).
eq(g(V1, Out),1,[],[Out = 0,V1 = 1]).
eq(encArg(V1, Out),0,[],[Out = 0,V1 = 0]).
eq(encArg(V1, Out),0,[encArg(V5, Ret0),encArg(V4, Ret1),encArg(V3, Ret2),f(Ret0, Ret1, Ret2, Ret3)],[Out = Ret3,V5 >= 0,V1 = 1 + V3 + V4 + V5,V3 >= 0,V4 >= 0]).
eq(encArg(V1, Out),0,[encArg(V6, Ret01),g(Ret01, Ret4)],[Out = Ret4,V1 = 1 + V6,V6 >= 0]).
eq(encArg(V1, Out),0,[b(Ret5)],[Out = Ret5,V1 = 2]).
eq(fun(V1, V, V2, Out),0,[encArg(V7, Ret02),encArg(V9, Ret11),encArg(V8, Ret21),f(Ret02, Ret11, Ret21, Ret6)],[Out = Ret6,V7 >= 0,V8 >= 0,V9 >= 0,V1 = V7,V = V9,V2 = V8]).
eq(fun1(V1, Out),0,[encArg(V10, Ret03),g(Ret03, Ret7)],[Out = Ret7,V10 >= 0,V1 = V10]).
eq(fun2(Out),0,[b(Ret8)],[Out = Ret8]).
eq(fun3(Out),0,[],[Out = 0]).
eq(b(Out),0,[],[Out = 1]).
eq(g(V1, Out),0,[],[Out = 1 + V11,V1 = V11,V11 >= 0]).
eq(encArg(V1, Out),0,[],[Out = 0,V12 >= 0,V1 = V12]).
eq(fun(V1, V, V2, Out),0,[],[Out = 0,V14 >= 0,V2 = V15,V13 >= 0,V1 = V14,V = V13,V15 >= 0]).
eq(fun1(V1, Out),0,[],[Out = 0,V16 >= 0,V1 = V16]).
eq(fun2(Out),0,[],[Out = 0]).
eq(b(Out),0,[],[Out = 0]).
eq(g(V1, Out),0,[],[Out = 0,V17 >= 0,V1 = V17]).
eq(f(V1, V, V2, Out),0,[],[Out = 0,V18 >= 0,V2 = V19,V20 >= 0,V1 = V18,V = V20,V19 >= 0]).
input_output_vars(b(Out),[],[Out]).
input_output_vars(f(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(g(V1,Out),[V1],[Out]).
input_output_vars(encArg(V1,Out),[V1],[Out]).
input_output_vars(fun(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(fun1(V1,Out),[V1],[Out]).
input_output_vars(fun2(Out),[],[Out]).
input_output_vars(fun3(Out),[],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. non_recursive  : [b/1]
1. recursive  : [f/4]
2. non_recursive  : [g/2]
3. recursive [non_tail,multiple] : [encArg/2]
4. non_recursive  : [fun/4]
5. non_recursive  : [fun1/2]
6. non_recursive  : [fun2/1]
7. non_recursive  : [fun3/1]
8. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into b/1
1. SCC is partially evaluated into f/4
2. SCC is partially evaluated into g/2
3. SCC is partially evaluated into encArg/2
4. SCC is partially evaluated into fun/4
5. SCC is partially evaluated into fun1/2
6. SCC is partially evaluated into fun2/1
7. SCC is completely evaluated into other SCCs
8. SCC is partially evaluated into start/3

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations b/1 
* CE 10 is refined into CE [27] 
* CE 9 is refined into CE [28] 
* CE 11 is refined into CE [29] 


### Cost equations --> "Loop" of b/1 
* CEs [27] --> Loop 15 
* CEs [28,29] --> Loop 16 

### Ranking functions of CR b(Out) 

#### Partial ranking functions of CR b(Out) 


### Specialization of cost equations f/4 
* CE 13 is refined into CE [30] 
* CE 12 is refined into CE [31] 


### Cost equations --> "Loop" of f/4 
* CEs [31] --> Loop 17 
* CEs [30] --> Loop 18 

### Ranking functions of CR f(V1,V,V2,Out) 

#### Partial ranking functions of CR f(V1,V,V2,Out) 


### Specialization of cost equations g/2 
* CE 15 is refined into CE [32] 
* CE 14 is refined into CE [33] 
* CE 16 is refined into CE [34] 


### Cost equations --> "Loop" of g/2 
* CEs [32] --> Loop 19 
* CEs [33,34] --> Loop 20 

### Ranking functions of CR g(V1,Out) 

#### Partial ranking functions of CR g(V1,Out) 


### Specialization of cost equations encArg/2 
* CE 17 is refined into CE [35] 
* CE 20 is refined into CE [36,37] 
* CE 19 is refined into CE [38,39] 
* CE 18 is refined into CE [40] 


### Cost equations --> "Loop" of encArg/2 
* CEs [40] --> Loop 21 
* CEs [39] --> Loop 22 
* CEs [38] --> Loop 23 
* CEs [37] --> Loop 24 
* CEs [35,36] --> Loop 25 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [21,22,23]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [21,22,23]:
  - RF of loop [21:1,21:2,21:3,22:1,23:1]:
    V1


### Specialization of cost equations fun/4 
* CE 21 is refined into CE [41,42,43,44,45,46,47,48] 
* CE 22 is refined into CE [49] 


### Cost equations --> "Loop" of fun/4 
* CEs [41,42,43,44,45,46,47,48,49] --> Loop 26 

### Ranking functions of CR fun(V1,V,V2,Out) 

#### Partial ranking functions of CR fun(V1,V,V2,Out) 


### Specialization of cost equations fun1/2 
* CE 23 is refined into CE [50,51,52,53] 
* CE 24 is refined into CE [54] 


### Cost equations --> "Loop" of fun1/2 
* CEs [51] --> Loop 27 
* CEs [53] --> Loop 28 
* CEs [50,52,54] --> Loop 29 

### Ranking functions of CR fun1(V1,Out) 

#### Partial ranking functions of CR fun1(V1,Out) 


### Specialization of cost equations fun2/1 
* CE 25 is refined into CE [55,56] 
* CE 26 is refined into CE [57] 


### Cost equations --> "Loop" of fun2/1 
* CEs [56] --> Loop 30 
* CEs [55,57] --> Loop 31 

### Ranking functions of CR fun2(Out) 

#### Partial ranking functions of CR fun2(Out) 


### Specialization of cost equations start/3 
* CE 1 is refined into CE [58,59] 
* CE 2 is refined into CE [60] 
* CE 3 is refined into CE [61,62] 
* CE 4 is refined into CE [63,64] 
* CE 5 is refined into CE [65] 
* CE 6 is refined into CE [66,67,68] 
* CE 7 is refined into CE [69,70] 
* CE 8 is refined into CE [71] 


### Cost equations --> "Loop" of start/3 
* CEs [58,59,60,61,62,63,64,65,66,67,68,69,70,71] --> Loop 32 

### Ranking functions of CR start(V1,V,V2) 

#### Partial ranking functions of CR start(V1,V,V2) 


Computing Bounds
=====================================

#### Cost of chains of b(Out):
* Chain [16]: 1
  with precondition: [Out=0] 

* Chain [15]: 0
  with precondition: [Out=1] 


#### Cost of chains of f(V1,V,V2,Out):
* Chain [18]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 

* Chain [17,18]: 1
  with precondition: [Out=0,V1+1=V,V1>=0,V2>=0] 


#### Cost of chains of g(V1,Out):
* Chain [20]: 1
  with precondition: [Out=0,V1>=0] 

* Chain [19]: 0
  with precondition: [V1+1=Out,V1>=0] 


#### Cost of chains of encArg(V1,Out):
* Chain [25]: 1
  with precondition: [Out=0,V1>=0] 

* Chain [24]: 0
  with precondition: [V1=2,Out=1] 

* Chain [multiple([21,22,23],[[25],[24]])]: 2*it(21)+1*it([25])+0
  Such that:it([25]) =< 2*V1+1
aux(1) =< V1
it(21) =< aux(1)

  with precondition: [V1>=1,Out>=0,V1>=Out] 


#### Cost of chains of fun(V1,V,V2,Out):
* Chain [26]: 4*s(5)+8*s(6)+4*s(8)+8*s(9)+4*s(11)+8*s(12)+4
  Such that:aux(2) =< V1
aux(3) =< 2*V1+1
aux(4) =< V
aux(5) =< 2*V+1
aux(6) =< V2
aux(7) =< 2*V2+1
s(5) =< aux(3)
s(8) =< aux(5)
s(11) =< aux(7)
s(12) =< aux(6)
s(9) =< aux(4)
s(6) =< aux(2)

  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 


#### Cost of chains of fun1(V1,Out):
* Chain [29]: 1*s(41)+2*s(42)+2
  Such that:s(40) =< V1
s(41) =< 2*V1+1
s(42) =< s(40)

  with precondition: [Out=0,V1>=0] 

* Chain [28]: 1
  with precondition: [Out=1,V1>=0] 

* Chain [27]: 1*s(44)+2*s(45)+0
  Such that:s(43) =< V1
s(44) =< 2*V1+1
s(45) =< s(43)

  with precondition: [V1>=1,Out>=1,V1+1>=Out] 


#### Cost of chains of fun2(Out):
* Chain [31]: 1
  with precondition: [Out=0] 

* Chain [30]: 0
  with precondition: [Out=1] 


#### Cost of chains of start(V1,V,V2):
* Chain [32]: 7*s(47)+14*s(48)+4*s(56)+4*s(57)+8*s(58)+8*s(59)+4
  Such that:s(51) =< V
s(52) =< 2*V+1
s(53) =< V2
s(54) =< 2*V2+1
aux(8) =< V1
aux(9) =< 2*V1+1
s(47) =< aux(9)
s(48) =< aux(8)
s(56) =< s(52)
s(57) =< s(54)
s(58) =< s(53)
s(59) =< s(51)

  with precondition: [] 


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [32] with precondition: [] 
    - Upper bound: nat(V1)*14+4+nat(V)*8+nat(V2)*8+nat(2*V1+1)*7+nat(2*V+1)*4+nat(2*V2+1)*4 
    - Complexity: n 

### Maximum cost of start(V1,V,V2): nat(V1)*14+4+nat(V)*8+nat(V2)*8+nat(2*V1+1)*7+nat(2*V+1)*4+nat(2*V2+1)*4 
Asymptotic class: n 
* Total analysis performed in 252 ms.


----------------------------------------

(16)
BOUNDS(1, n^1)